So next, I would like to describe the action of the Steenrod squares in a more general context (which is pure homological algebra). This more general approach to Steenrod operations seems to be due to Peter May, but I learned it from these lecture notes by Jacob Lurie.

As I said earlier, the Steenrod squares will act on the cohomology of any ${E_\infty}$-algebra. What is an ${E_\infty}$-algebra? It’s supposed to be a dga with a multiplication law that is coherent and associative up to coherent homotopy. Or, equivalently, an algebra over an ${E_\infty}$-operad in chain complexes. Unfortunately, I don’t know too many simple ones. For instance, the standard ${E_\infty}$-operad in spaces, the infinite little cubes operad, is huge: the individual terms have to be contractible spaces large enough to admit a free action of the symmetric group.

But we’ll need less structure than an ${E_\infty}$-algebra structure. We’ll need a multiplication law on a chain complex ${X}$ which is commutative up to coherent homotopy, but without quite all the coherence that one would need for an ${E_\infty}$-algebra.

What should this be? Well, a multiplication law would be a map ${X \otimes X \rightarrow X}$, and to say that it is commutative would be to say that it is ${\Sigma_2}$-equivariant where ${\Sigma_2}$ acts by permutation on ${X \otimes X}$ and trivially on ${X}$. In other words, it would be a map ${(X \otimes X)_{\Sigma_2} \rightarrow X}$ where ${(X \otimes X)_{\Sigma_2}}$ denotes the coinvariants. This is generally going to be too strong: most of the homotopy commutative multiplication laws we deal with will not be rectifiable to strictly commutative ones.

Instead, we’ll use maps not out of the coinvariants of ${(X \otimes X)_{\Sigma_2}}$, but out of the homotopy coinvariants. To get this, one first tensors ${X \otimes X}$ with a complex of free ${\Sigma_2}$-modules which is contractible as a complex of vector spaces, and then takes the coinvariants.

More generally:

Definition 1 If ${Y}$ is a ${\Sigma_2}$-equivariant complex of ${\mathbb{F}_2}$-vector spaces, then we define the homotopy coinvariants ${(h Y)_{\Sigma_2}}$ as the coinvariants of ${(Y \otimes E \Sigma_2)_{\Sigma_2}}$. Here ${E \Sigma_2}$ is any acyclic resolution of the complex ${\mathbb{F}_2[0]}$ (just ${\mathbb{F}_2}$ in degree zero, nothing elsewhere) consisting of free ${\mathbb{F}_2[\Sigma_2]}$-modules.

I’ve been reading Milnor’s paper “The Steenrod algebra and its dual,” and want to talk a little about it today. The starting point of this story is the theory of cohomology operations. Given a cohomology theory ${h^*}$ on spaces (or just CW complexes; one can always Kan extend to all spaces), one can consider cohomology operations on ${h^*}$. Most interesting for our purposes are the stable cohomology operations.

A stable cohomology operation of degree ${k}$ will be a collection of homomorphisms ${h^m(X) \rightarrow h^{m+k}(X)}$ for each ${m}$, which are natural in the space ${X}$, and which commute with the suspension isomorphisms. If we think of ${h^*}$ as represented by a spectrum ${E}$, so that ${h^*(X) = [X, E]}$ is a representable functor (in the stable homotopy category), then a stable cohomology operation comes from a homotopy class of maps ${E \rightarrow E}$ of degree ${k}$.

A stable cohomology operation is additive, because it comes from a spectrum map, and the stable homotopy category is additive. Moreover, the set of all stable cohomology operations becomes a graded ring under composition. It is equivalently the graded ring ${[E, E]}$.

The case where ${E}$ is an Eilenberg-MacLane spectrum, and ${h^*}$ ordinary cohomology, is itself pretty interesting. First off, one has to work in finite characteristic—in characteristic zero, there are no nontrivial stable cohomology operations. In fact, the only (possibly unstable) natural transformations ${H^*(\cdot, \mathbb{Q}) \rightarrow H^*(\cdot, \mathbb{Q})}$ come from taking iterated cup products because ${H^*(K(\mathbb{Q}, n))}$ can be computed, via the spectral sequence, to be a free graded-commutative algebra over ${\mathbb{Q}}$ generated by the universal element. These aren’t stable, so the only stable one has to be zero. So we will work with coefficients ${\mathbb{Z}/p}$ for ${p}$ a prime.

Here the algebra of stable cohomology operations is known and has been known since the 1950’s; it’s called the Steenrod algebra ${\mathcal{A}^*}$. In fact, all unstablecohomology operations are themselves known. Let me state the result for ${p=2}$.

Steenrod had constructed squaring operations

$\displaystyle \mathrm{Sq}^i: H^*(\cdot, \mathbb{Z}/2) \rightarrow H^{* +i}(\mathbb{Z}/2) .$

These are natural transformations, which have the following properties:

1. ${\mathrm{Sq}^0}$ is the identity operation.
2. ${\mathrm{Sq}^i}$ on a cohomology class ${x}$ of dimension ${n}$ vanishes for ${i > n}$. For ${i = n}$, ${\mathrm{Sq}^i}$ acts by the cup square on ${x}$.
3. The Steenrod squares behave well with respect to the cohomology cross (and thus cup) product: ${\mathrm{Sq}^i(a \times b) = \sum_{j + k = i} \mathrm{Sq}^j a \times \mathrm{Sq}^k b}$.
4. ${\mathrm{Sq}^1}$ is the Bockstein connecting homomorphism associated to the short exact sequence ${0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2 \rightarrow 0}$.
5. ${\mathrm{Sq}^i}$ commutes with suspension (and thus is a homomorphism). (more…)